LECTURE 4
Syntax
SPECIFYING SYNTAX
Programming languages must be very well defined – there’s no room for ambiguity.
Language designers must use formal syntactic and semantic notation to specify the
rules of a language.
In this lecture, we will focus on how syntax is specified.
SPECIFYING SYNTAX
We know from the previous lecture that the front-end of the compiler has three main
phases:
• Scanning
• Parsing
Syntax Verification
• Semantic Analysis
SPECIFYING SYNTAX
• Scanning
• Identifies the valid tokens, the basic building blocks, within a program.
• Parsing
• Identifies the valid patterns of tokens, or constructs.
So how do we specify what a valid token is? Or what constitutes a valid construct?
REGULAR EXPRESSIONS
Tokens can be constructed from regular characters using just three rules:
1. Concatenation.
2. Alternation (choice among a finite set of alternatives).
3. Kleene Closure (arbitrary repetition).
Any set of strings that can be defined by these three rules is a regular set.
Regular sets are generated by regular expressions.
REGULAR EXPRESSIONS
Formally, all of the following are valid regular expressions (let R and S be regular
expressions and let Σ be a finite set of symbols):
• The empty set.
• The set containing the empty string 𝜖.
• The set containing a single literal character 𝛼 from the alphabet Σ.
• Concatenation: RS is the set of strings obtained by concatenation of one string from
R with a string from S.
• Alternation: R|S describes the union of R and S.
• Kleene Closure: R* is the set of strings that can be obtained by concatenating any
number of strings from R.
REGULAR EXPRESSIONS
You can either use parentheses to avoid ambiguity or assume Kleene star has the
highest priority, followed by concatenation then alternation.
Examples:
• a* = {𝜖, a, aa, aaa, aaaa, aaaaa, …}
• a | b* = {𝜖, a, b, bb, bbb, bbbb, …}
• (ab)* = {𝜖, ab, abab, ababab, abababab, …}
• (a|b)* = {𝜖, a, b, aa, ab, ba, bb, aaa, aab, …}
REGULAR EXPRESSIONS
Write a regular expression for each of the following:
• Zero or more c’s followed by a single a or a single b.
• Binary strings starting and ending with 1.
• Binary strings containing at least 3 1’s.
REGULAR EXPRESSIONS
Write a regular expression for each of the following:
• Zero or more c’s followed by a single a or a single b.
• c*(a|b)
• Binary strings starting and ending with 1.
• 1|1(0|1)*1
• Binary strings containing at least 3 1’s.
• 0*10*10*1(0|1)*
REGULAR EXPRESSIONS
Let’s look at a more practical example. Say we want to write a regular expression to
identify valid numbers.
Some things to consider:
•
•
•
•
Numbers can be any number of digits long, but must not start with 0.
Numbers can be positive or negative.
Numbers can be integers or real.
Numbers can be represented by scientific notation (i.e. 2.9e8).
REGULAR EXPRESSIONS
number  integer | real
integer  non_zero_digit digit*
real  integer exponent | decimal (exponent | 𝜖)
decimal  (non_zero_digit digit*)* ( . digit | digit . ) digit*
exponent  (e | E) (+ | - | 𝜖) integer
digit  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
non_zero_digit  1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
REGULAR EXPRESSIONS
So our number tokens are well-defined by the number symbol, which makes use of the
other symbols to build larger expressions.
Any valid pattern generated by expanding out the number symbol is a valid number.
Note: while our rules build upon one another, no symbol is defined in terms of itself,
even indirectly.
number  integer | real
integer  non_zero_digit digit*
real  integer exponent | decimal (exponent | 𝜖)
decimal  (non_zero_digit digit*)* ( . digit | digit . ) digit*
exponent  (e | E) (+ | - | 𝜖) integer
digit  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
non_zero_digit  1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
CONTEXT-FREE GRAMMARS
We can completely define our tokens in terms of regular expressions, but more
complicated constructs necessitate the ability to self-reference.
This self-referencing ability takes the form of recursion.
The set of strings that can be defined by adding recursion to regular expressions is
known as a Context-Free Language.
Context-Free Languages are generated by Context-Free Grammars.
CONTEXT-FREE GRAMMARS
We’ve seen a little bit of context-free grammars, but let’s flesh out the details.
Context-free grammars are composed of rules known as productions.
Each production has left-hand side symbols known as non-terminals, or variables.
On the right-hand side, a production may contain terminals (tokens) or other nonterminals.
One of the non-terminals is named the start symbol.
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
This notation is known as Backus-Naur Form.
DERIVATIONS
So, how do we use the context-free grammar to generate syntactically valid strings of
terminals (or tokens)?
1. Begin with the start symbol.
2. Choose a production with the start symbol on the left side.
3. Replace the start symbol with the right side of the chosen production.
4. Choose a non-terminal A in the resulting string.
5. Replace A with the right side of a production whose left side is A.
6. Repeat 4 and 5 until no non-terminals remain.
DERIVATIONS
Let’s do a practice derivation with our grammar.
We’ll derive the string “(base1 + base2) * height/2”.
The start symbol is expr.
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
Each string of symbols in the steps of the
derivation is called a sentential form.
The final sentential form is known as the yield.
DERIVATIONS
To save a little bit of room, we can write:
expr
* ( id + id ) * id / number
“derives after zero or
more replacements”
Note that in this derivation, we replaced the right-hand side consistently,
leading to a right-most derivation. There are alternative derivation methods.
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
op
expr
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
op
expr
expr op expr
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
op
expr
expr op expr
num
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
op
expr
expr op expr
/
num
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
op
expr
expr op expr
id
/
num
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
op
expr
*
expr op expr
id
/
num
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
(
expr
)
op
expr
*
expr op expr
id
/
num
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
(
expr
)
expr
op
expr
op
expr
*
expr op expr
id
/
num
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
(
expr
)
expr
op
expr
id
op
expr
*
expr op expr
id
/
num
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
(
expr
)
expr
op
expr
+
id
op
expr
*
expr op expr
id
/
num
PARSE TREES FROM DERIVATIONS
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr op expr
expr op expr op expr
expr op expr op number
expr op expr / number
expr op id / number
expr * id / number
( expr ) * id / number
( expr op expr ) * id / number
( expr op id ) * id / number
( expr + id ) * id / number
( id + id ) * id / number
expr
expr
(
expr
)
expr
op
expr
id
+
id
op
expr
*
expr op expr
id
/
num
PARSE TREES FROM DERIVATIONS
Consider the following: “length * width * height”
From our grammar, we can generate two equally acceptable parse trees.
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr
expr
op
id
*
op
expr *
id
expr
expr
expr
op
id
id
*
expr
expr
op
expr
id
*
id
AMBIGUOUS DERIVATIONS
Consider the following: “length * width * height”
From our grammar, we can generate two equally acceptable parse trees.
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
expr
expr
op
id
*
op
expr *
id
expr
expr
expr
op
id
id
*
expr
expr
op
expr
id
*
id
Grammars that allow more than one parse tree for the same string are said to be ambiguous.
Parsers must, in practice, generate special rules for disambiguation.
AMBIGUOUS DERIVATIONS
Context-free grammars can be structured such that derivations are more efficient for
the compiler.
Take the example of arithmetic expressions. In most languages, multiplication and
division take precedence over addition and subtraction. Also, associativity tells us that
operators group left to right.
We could allow ambiguous derivations and let the compiler sort out the precedence
later or we could just build it into the structure of the parse tree.
AMBIGUOUS DERIVATIONS
Previously, we had:
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
Building in associativity and operator precedence:
expr  term | expr add_op term
term  factor | term mult_op factor
factor  id | number | - factor | ( expr )
add_op  + | mult_op  * | /
AMBIGUOUS DERIVATIONS
Previously, we had:
Example: 3 + 4 * 5
expr  id | number | - expr | ( expr ) | expr op expr
op  + | - | * | /
expr
Building in associativity and operator precedence:
expr
add_op
expr  term | expr add_op term
term  factor | term mult_op factor
factor  id | number | - factor | ( expr )
add_op  + | mult_op  * | /
term
+
term
term
mult_op
factor
factor
factor
*
number
number
number
NEXT LECTURE
Scanning
Finite Automata: NFAs and DFAs
Implementing a Scanner